3F4 Pulse Amplitude Modulation (PAM) Dr. I. J. Wassell

Introduction The purpose of the modulator is to convert discrete amplitude serial symbols (bits in a binary system) a k to analogue output pulses which are sent over the channel. The demodulator reverses this process ModulatorChannelDemodulator Serial data symbols akak ‘analogue’ channel pulses Recovered data symbols

Introduction Possible approaches include –Pulse width modulation (PWM) –Pulse position modulation (PPM) –Pulse amplitude modulation (PAM) We will only be considering PAM in these lectures

PAM PAM is a general signalling technique whereby pulse amplitude is used to convey the message For example, the PAM pulses could be the sampled amplitude values of an analogue signal We are interested in digital PAM, where the pulse amplitudes are constrained to chosen from a specific alphabet at the transmitter

PAM Scheme H C (  ) h C (t) Symbol clock H T (  ) h T (t) Noise N(  ) Channel + Pulse generator akak Transmit filter Receive filter H R (  ), h R (t) Data slicer Recovered symbols Recovered clock Modulator Demodulator

PAM In binary PAM, each symbol a k takes only two values, say {A 1 and A 2 } In a multilevel, i.e., M-ary system, symbols may take M values {A 1, A 2,... A M } Signalling period, T Each transmitted pulse is given by Where h T (t) is the time domain pulse shape

PAM To generate the PAM output signal, we may choose to represent the input to the transmit filter h T (t) as a train of weighted impulse functions Consequently, the filter output x(t) is a train of pulses, each with the required shape h T (t)

PAM Filtering of impulse train in transmit filter Transmit Filter

PAM Clearly not a practical technique so –Use a practical input pulse shape, then filter to realise the desired output pulse shape –Store a sampled pulse shape in a ROM and read out through a D/A converter The transmitted signal x(t) passes through the channel H C (  ) and the receive filter H R (  ). The overall frequency response is H(  ) = H T (  ) H C (  ) H R (  )

PAM Hence the signal at the receiver filter output is Where h(t) is the inverse Fourier transform of H(  ) and v(t) is the noise signal at the receive filter output Data detection is now performed by the Data Slicer

PAM- Data Detection Sampling y(t), usually at the optimum instant t=nT+t d when the pulse magnitude is the greatest yields Where v n =v(nT+t d ) is the sampled noise and t d is the time delay required for optimum sampling y n is then compared with threshold(s) to determine the recovered data symbols

PAM- Data Detection Data Slicer decision threshold = 0V  Signal at data slicer input, y(t) Sample clock Sampled signal, y n = y(nT+t d ) Ideal sample instants at t = nT+t d  TX data TX symbol, a k ‘1’‘0’ ‘1’‘0’ +A-A +A-A Detected data‘1’‘0’ ‘1’‘0’  tdtd

Synchronisation We need to derive an accurate clock signal at the receiver in order that y(t) may be sampled at the correct instant Such a signal may be available directly (usually not because of the waste involved in sending a signal with no information content) Usually, the sample clock has to be derived directly from the received signal.

Synchronisation The ability to extract a symbol timing clock usually depends upon the presence of transitions or zero crossings in the received signal. Line coding aims to raise the number of such occurrences to help the extraction process. Unfortunately, simple line coding schemes often do not give rise to transitions when long runs of constant symbols are received.

Synchronisation Some line coding schemes give rise to a spectral component at the symbol rate A BPF or PLL can be used to extract this component directly Sometimes the received data has to be non- linearly processed eg, squaring, to yield a component of the correct frequency.

Intersymbol Interference If the system impulse response h(t) extends over more than 1 symbol period, symbols become smeared into adjacent symbol periods Known as intersymbol interference (ISI) The signal at the slicer input may be rewritten as –The first term depends only on the current symbol a n –The summation is an interference term which depends upon the surrounding symbols

Intersymbol Interference Example –Response h(t) is Resistor-Capacitor (R-C) first order arrangement- Bit duration is T For this example we will assume that a binary ‘0’ is sent as 0V. Time (bit periods)  amplitude   Time (bit periods)  amplitude   Modulator inputSlicer input Binary ‘1’

Intersymbol Interference The received pulse at the slicer now extends over 4 bit periods giving rise to ISI. The actual received signal is the superposition of the individual pulses time (bit periods)  amplitude   ‘1’ ‘0’ ‘1’‘0’ ‘1’

Intersymbol Interference For the assumed data the signal at the slicer input is, Clearly the ease in making decisions is data dependant time (bit periods)  amplitude   Note non-zero values at ideal sample instants corresponding with the transmission of binary ‘0’s ‘1’ ‘0’ ‘1’‘0’ ‘1’ Decision threshold

Intersymbol Interference Matlab generated plot showing pulse superposition (accurately) Decision threshold time (bit periods) Received signal Individual pulses amplitude

Intersymbol Interference Sending a longer data sequence yields the following received waveform at the slicer input Decision threshold (Also showing individual pulses)

Eye Diagrams Worst case error performance in noise can be obtained by calculating the worst case ISI over all possible combinations of input symbols. A convenient way of measuring ISI is the eye diagram Practically, this is done by displaying y(t) on a scope, which is triggered using the symbol clock The overlaid pulses from all the different symbol periods will lead to a criss-crossed display, with an eye in the middle

Example R-C response Eye Diagram Decision threshold Optimum sample instant h = eye height h

Eye Diagrams The size of the eye opening, h (eye height) determines the probability of making incorrect decisions The instant at which the max eye opening occurs gives the sampling time t d The width of the eye indicates the resilience to symbol timing errors For M-ary transmission, there will be M-1 eyes

Eye Diagrams The generation of a representative eye assumes the use of random data symbols For simple channel pulse shapes with binary symbols, the eye diagram may be constructed manually by finding the worst case ‘1’ and worst case ‘0’ and superimposing the two

Nyquist Pulse Shaping It is possible to eliminate ISI at the sampling instants by ensuring that the received pulses satisfy the Nyquist pulse shaping criterion We will assume that t d =0, so the slicer input is If the received pulse is such that

Nyquist Pulse Shaping Then and so ISI is avoided This condition is only achieved if That is the pulse spectrum, repeated at intervals of the symbol rate sums to a constant value T for all frequencies

Nyquist Pulse Shaping  T  f  H(f) T f

Why? Sample h(t) with a train of  pulses at times kT Consequently the spectrum of h s (t) is Remember for zero ISI

Why? Consequently h s (t)=  (t) The spectrum of  (t)=1, therefore Substituting f=  /2  gives the Nyquist pulse shaping criterion

Nyquist Pulse Shaping  T  f No pulse bandwidth less than 1/2T can satisfy the criterion, eg, Clearly, the repeated spectra do not sum to a constant value

Nyquist Pulse Shaping The minimum bandwidth pulse spectrum H(f), ie, a rectangular spectral shape, has a sinc pulse response in the time domain, The sinc pulse shape is very sensitive to errors in the sample timing, owing to its low rate of sidelobe decay

Nyquist Pulse Shaping Hard to design practical ‘brick-wall’ filters, consequently filters with smooth spectral roll-off are preferred Pulses may take values for t<0 (ie non- causal). No problem in a practical system because delays can be introduced to enable approximate realisation.

Causal Response Non-causal response T = 1 s Causal response T = 1s Delay, t d = 10s

Raised Cosine (RC) Fall-Off Pulse Shaping Practically important pulse shapes which satisfy the criterion are those with Raised Cosine (RC) roll-off The pulse spectrum is given by With, 0<  <1/2T

RC Pulse Shaping The general RC function is as follows, H(f) f (Hz) T 0

RC Pulse Shaping The corresponding time domain pulse shape is given by, Now  allows a trade-off between bandwidth and the pulse decay rate Sometimes  is normalised as follows,

RC Pulse Shaping With  =0 (i.e., x = 0) the spectrum of the filter is rectangular and the time domain response is a sinc pulse, that is, The time domain pulse has zero crossings at intervals of nT as desired (See plots for x = 0).

RC Pulse Shaping With  =(1/2T), (i.e., x = 1) the spectrum of the filter is full RC and the time domain response is a pulse with low sidelobe levels, that is, The time domain pulse has zero crossings at intervals of nT/2, with the exception at T/2 where there is no zero crossing. See plots for x = 1.

RC Pulse Shaping Normalised Spectrum H(f)/TPulse Shape h(t) x  x  x  f *Tt/T

RC Pulse Shaping- Example 1 Eye diagram Pulse shape and received signal, x = 0 (  = 0)

RC Pulse Shaping- Example 2 Eye diagram Pulse shape and received signal, x = 1 (  = 1/2T)

RC Pulse Shaping- Example The much wider eye opening for x = 1 gives a much greater tolerance to inaccurate sample clock timing The penalty is the much wider transmitted bandwidth

Probability of Error In the presence of noise, there will be a finite chance of decision errors at the slicer output The smaller the eye, the higher the chance that the noise will cause an error. For a binary system a transmitted ‘1’ could be detected as a ‘0’ and vice-versa In a PAM system, the probability of error is, P e =Pr{A received symbol is incorrectly detected} For a binary system, P e is known as the bit error probability, or the bit error rate (BER)

BER The received signal at the slicer is Where V i is the received signal voltage and V i =V o for a transmitted ‘0’ or V i =V 1 for a transmitted ‘1’ With zero ISI and an overall unity gain, V i =a n, the current transmitted binary symbol Suppose the noise is Gaussian, with zero mean and variance

BER Where f(v n ) denotes the probability density function (pdf), that is, and

BER dx vnvn f(v n ) ba 0

BER The slicer detects the received signal using a threshold voltage V T For a binary system the decision is Decide ‘1’ if y n V T Decide ‘0’ if y n <V T For equiprobable symbols, the optimum threshold is in the centre of V 0 and V 1, ie V T =(V 0 +V 1 )/2

BER ynyn f(y n |‘0’ sent) VTVT 0f(y n |‘1’ sent) P(error|‘0’)P(error|‘1’) V1V1 V0V0

BER The probability of error for a binary system can be written as: P e =Pr(‘0’sent and error occurs)+Pr(‘1’sent and error occurs) For ‘0’ sent: an error occurs when y n V T –let v n =y n -V o, so when y n =V o, v n =0 and when y n =V T, v n =V T -V o. –So equivalently, we get an error when v n V T -V 0

BER ynyn f(y n |‘0’ sent) VTVT 0 P(error|‘0’) V0V0 vnvn f(v n ) V T - V 0 0 P(error|‘0’)

BER The Q function is one of a number of tabulated functions for the Gaussian cumulative distribution function (cdf) ie the integral of the Gaussian pdf. Where,

BER Similarly for ‘1’ sent: an error occurs when y n <V T –let v n =y n -V 1, so when y n =V 1, v n =0 and when y n =V T, v n =V T -V 1. –So equivalently, we get an error when v n < V T -V 1

BER ynyn VTVT 0 f(y n |‘1’ sent) P(error|‘1’) V1V1 V T -V 1 vnvn 0 f(v n ) P(error|‘1’) -(V T -V 1 ) V 1 -V T

BER Hence the total error probability is P e =Pr(‘0’sent and error occurs)+Pr(‘1’sent and error occurs) Where P o is the probability that a ‘0’ was sent and P 1 is the probability that a ‘1’ was sent For P o =P 1 =0.5, the min error rate is obtained when,

BER Consequently, Notes: –Q(.) is a monotonically decreasing function of its argument, hence the BER falls as h increases –For received pulses satisfying Nyquist criterion, ie zero ISI, V o =A o and V 1 =A 1. Assuming unity overall gain. –More complex with ISI. Worst case performance if h is taken to be the eye opening

BER Example The received pulse h(t) in response to a single transmitted binary ‘1’ is as shown, Where, h(0) = 0, h(T) = 0.3, h(2T) = 1, h(3T) = 0, h(4T) = -0.2, h(5T) =0 Bit period = T t         h(t) (V)

BER Example What is the worst case BER if a ‘1’ is received as h(t) and a ‘0’ as -h(t) (this is known as a polar binary scheme)? Assume the data are equally likely to be ‘0’ and ‘1’ and that the optimum threshold (OV) is used at the slicer. By inspection, the pulse has only 2 non-zero amplitude values (at T and 4T) away from the ideal sample point (at 2T).

BER Example Consequently the worst case ‘1’ occurs when the data bits conspire to give negative non-zero pulse amplitudes at the sampling instant. The worst case ‘1’ eye opening is thus, = 0.5 as indicated in the following diagram.

BER Example The indicated data gives rise to the worst case ‘1’ eye opening. Don’t care about data marked ‘X’ as their pulses are zero at the indicated sample instant

BER Example Similarly the worst case ‘0’ eye opening is = -0.5 So, worst case eye opening h = 0.5-(-0.5) = 1V Giving the BER as, Where  v is the rms noise at the slicer input

Summary For PAM systems we have –Looked at ISI and its assessment using eye diagrams –Nyquist pulse shaping to eliminate ISI at the optimum sampling instants –Seen how to calculate the worst case BER in the presence of Gaussian noise and ISI